In a small saucepan, bring the sugar, lime zest and 1 cup water to a simmer over medium heat, stirring constantly until the sugar dissolves, about 6 minutes. Remove from the heat and stir in the mint leaves. Let cool for 30 minutes.

Strain the lime syrup through a sieve into a pitcher. Add the cucumber and lime juice; refrigerate for at least 1 hour.

Just before serving, add the sparkling water. Serve over ice and garnish with mint.